In a typical prior machining system of this general type, a beam from a pulsed laser is used to form a relatively large number of holes in a workpiece. Each laser pulse or series of pulses forms an individual hole. Between pulses (or series of pulses) the workpiece is moved mechanically to a new position relative to the beam to make a new hole, or the machining head directing the beam is moved while the workpiece stay fixed. In either case, the system requires a relatively long time (relative to the pulse repetition rate of the laser) for the workpiece or the machining head to be physically moved from alignment with one machining location on the workpiece to alignment with the next such location. Especially if a large number of holes spread out over an extensive workpiece surface are to be drilled, the total time required for the repeated mechanical realignment of the beam and the workpiece is excessive, resulting in a low throughput and high manufacturing costs.